It's still difficult to know what you did because you're providing such sketchy information in terms of quantities, sequences, and times. Also, there's a terminology problem. But, I might be able to provide some insight.
First, the ice cream base: I don't think one egg is enough to take 2-1/4 cups cream to a nappe consistency, so your expectation that the base would "set up" as a true custard may have been misplaced. Ice creams don't necessarily need a cooked-custard base to setup though. I'm surprised freezing the base in the churn didn't solve the problem for you. What happened when you tried to freeze it?
Second, the tart: Since you provided the source for the recipe, I was able to find it and check the sequence. I'm surprised that so much corn starch didn't thicken the milk up, let alone the eggs. By the way, the eggs were not supposed to be "tempered." The recipe specified that they were introduced directly into to the warm milk, which was then to be cooked and thickened.
The recipe looked bullet proof to me. If it didn't work, there's a major problem in your execution or something very strange with one of the ingredients. But with this little information I can't pinpoint it. On the basis of the ingredient list alone, perhaps you added the cream to the custard, rather than reserving it as a topping. You mentioned not using the "spoon" test, so perhaps you didn't cook the custard long enough. The recipe specified 3 minutes, and maybe you stopped after 3 minutes -- which might not have been enough. A good rule is that it's generally better to rely on visual and other sensory tests than on measures of time.